Exercise 10.16

Proof.

We can write a cubic form on $P_1({𝔽}_2)$ under the form

Thus there are $15$ such cubic forms.

This small Sage program computes the set of non trivial solutions for each of these forms

F2 = GF(2)
R.<x0,x1>= F2[]
l = [a*x0^3 + b * x0^2 * x1 + c * x0 * x1^2 + d * x1^3
      for a in F2 for b in F2 for c in F2 for d in F2
      if not [a,b,c,d] == [0,0,0,0]]
l

$[x_1^3,x_0{x}_1^2,x_0{x}_1^2+x_1^3,x_0^2{x}_1,x_0^2{x}_1+x_1^3,x_0^2{x}_1+x_0{x}_1^2,x_0^2{x}_1+x_0{x}_1^2+x_1^3,x_0^3,x_0^3+x_1^3,x_0^3+x_0{x}_1^2,x_0^3+x_0{x}_1^2+x_1^3,x_0^3+x_0^2{x}_1,x_0^3+x_0^2{x}_1+x_1^3,x_0^3+x_0^2{x}_1+x_0{x}_1^2,x_0^3+x_0^2{x}_1+x_0{x}_1^2+x_1^3]$

for f in l:
    S = []
    for x in F2:
        for y in F2:
            if [x,y] != [0,0] and f.subs(x0=x,x1=y) == 0:
                S.append([x,y])
    print f,  ’ : ’, S

x1^3   :  [[1, 0]]
x0*x1^2   :  [[0, 1], [1, 0]]
x0*x1^2 + x1^3   :  [[1, 0], [1, 1]]
x0^2*x1   :  [[0, 1], [1, 0]]
x0^2*x1 + x1^3   :  [[1, 0], [1, 1]]
x0^2*x1 + x0*x1^2   :  [[0, 1], [1, 0], [1, 1]]
x0^2*x1 + x0*x1^2 + x1^3   :  [[1, 0]]
x0^3   :  [[0, 1]]
x0^3 + x1^3   :  [[1, 1]]
x0^3 + x0*x1^2   :  [[0, 1], [1, 1]]
x0^3 + x0*x1^2 + x1^3   :  []
x0^3 + x0^2*x1   :  [[0, 1], [1, 1]]
x0^3 + x0^2*x1 + x1^3   :  []
x0^3 + x0^2*x1 + x0*x1^2   :  [[0, 1]]
x0^3 + x0^2*x1 + x0*x1^2 + x1^3   :  [[1, 1]]

This shows that two cubics forms have no non trivial solutions. We verify this for the form $x_0^3+x_0{x}_1^2+x_1^3$ :

So the sentence is false.

With three variables $x_0,x_1,x_2$, there are $1023$ cubics forms. A similar program gives among them the form

which has no non trivial zero:

The Chevalley’s Theorem shows that with 4 (or more) variables $x_0,x_1,x_2,x_3$, every cubic form has non trivial solutions. □